INDEX 747

critical

exponent, 246, 513, 553, 577,

681–688, 696

point, 117, 456, 503–508

value, 504

curvature, 213

mean, 459, 525

principal curvatures, 228

cutoﬀ function, 330

d’Alembert’s formula, 68, 69, 78, 81, 89

Deformation Theorem, 505, 508

DeGiorgi–Moser–Nash estimates, 489

degree, 523

Derrick–Pohozaev identity, 556, 577

diﬀerence quotients, 293–295, 331, 337,

475, 486, 667, 677

and weak derivatives, 294

deﬁnition of, 294

diﬀerential equations

in Banach spaces, 436

in Hilbert spaces, 506, 567

diﬀusion, 212, 315

and transport, 212–216

equation, 4, 44

DiPerna, R., 166

Dirac measure, 25, 35, 48, 212, 213

Dirichlet’s

boundary condition, 314

energy functional, 42, 525, 526

principle, 42, 458

generalized, 458

dispersion, 177

dissipation, 177

divergence form PDE, 314, 374, 529

domain of dependence

generalized wave equation, 418

nonlinear wave equation, 662–664

wave equation, 83

domain variations, 371, 513–521, 716

Dominated Convergence Theorem, 133,

153, 223, 435, 476, 548, 650, 674,

735

doubling variables, 589, 652

Douglis, A., 166

duality of Hamiltonian and Lagrangian,

121

dual space, 725

of H0

1,

301

dual variational principle, 250, 525

Duhamel’s principle, 49, 80

dynamic programming, 11, 592, 594,

607, 608

Egoroﬀ’s Theorem, 470, 734

eigenfunction, 169, 325

basis, 357

eigenvalue, 11, 169, 325, 356, 618, 620,

725

nonsymmetric elliptic operator,

362–367

principal, 11, 358–367, 448, 549, 576

symmetric elliptic operator, 356–362

eigenvector, 356, 620, 725

left, 618

right, 618

eikonal equation, 5, 93, 606

generalized, 418

elasticity

linear, 6, 89

nonlinear, 481, 502

ellipticity, 315

elliptic operator

fully nonlinear, 608

linear, 314

divergence form, 314, 315, 329, 356,

367

nondivergence form, 314, 346, 362

quasilinear, 485, 543

elliptic regularization, 523

Emden–Fowler transformation, 246

energy decay

wave equation, 521–522

energy estimates

quasilinear elliptic equation, 532

second-order elliptic equation, 319

second-order hyperbolic equation, 404

second-order parabolic equation, 378

symmetric hyperbolic system, 428

energy flux, 663, 679, 682

energy methods

heat equation, 62–65

Laplace’s equation, 41–43

wave equation, 82–84

entropy/entropy-flux pair, 648, 651,

657, 658

entropy condition, 12, 142, 148, 633, 643

condition E, 658

Lax’s, 633

Liu’s, 646

entropy solution

single conservation law, 149, 165, 658

system, 649, 652